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| ON THE RELATIVE POSITION OF LIMITCYCLES FOR THE EQUATIONOF TYPE $\[{(II)_{l = 0}}\]$ |
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Citation： |
Liang Zhaojun.ON THE RELATIVE POSITION OF LIMITCYCLES FOR THE EQUATIONOF TYPE $\[{(II)_{l = 0}}\]$[J].Chinese Annals of Mathematics B,1984,5(1):37~42 |
| Page view： 730
Net amount： 914 |
Authors： |
Liang Zhaojun; |
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| Abstract： |
In this paper, we consider the relative position of limit cycles for the system
$$\[\begin{array}{*{20}{c}}
{\frac{{dx}}{{dt}} = \delta x - y + mxy - {y^2}}\{\frac{{dy}}{{dt}} = x + a{x^2}}
\end{array}\]$$
under the condition
$$\[a < 0,0 < \delta \le m,m \le \frac{1}{a} - a\]$$
The main result is as follows:
(i)Under Condition (2), if $\[\delta = \frac{m}{2} + \frac{{{m^2}}}{{4a}} \equiv {\delta _0}\]$, then system $\[{(1)_{{\delta _0}}}\] $ has no limit cycles and
on singular closed trajectory through a saddle point in the whole plane,
(ii)Under condition (2), the foci 0 and R' cannot be surrounded by the limit cycles of system (1) simultaneously. |
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